Conversational query rewriting (CQR) addresses context dependence in conversational search by rewriting each user query into a standalone form. Recent approaches leverage reinforcement learning (RL) to directly optimize retrieval effectiveness; however, they typically rely on a single rewrite, which struggles to accommodate the divergent preferences of sparse and dense retrievers and often suffers from conflicting optimization signals. We propose DVCQR, a Dual-View CQR framework that explicitly generates two complementary rewrites for each query: a sparse-view rewrite that emphasizes distinctive lexical anchors, and a dense-view rewrite that captures complete semantic constraints. Both rewrites are produced in a single pass via a structured reasoning process. To further mitigate objective conflicts, we introduce a stage-wise RL strategy that sequentially aligns the sparse and dense views with their corresponding retrievers using rank-based feedback. Extensive experiments on four benchmarks (TopiOCQA, QReCC, CAsT-19, and CAsT-20) demonstrate that DVCQR consistently outperforms state-of-the-art methods on most metrics under both sparse and dense retrieval settings, validating the effectiveness of dual-view rewriting and stage-wise retriever alignment.